On the achievable rate of bandlimited continuous-time AWGN channels with 1-bit output quantization

We consider a continuous-time bandlimited additive white Gaussian noise channel with 1-bit output quantization. On such a channel the information is carried by the temporal distances of the zero-crossings of the transmit signal. The set of input signals is constrained by the bandwidth of the channel and an average power constraint. We derive a lower bound on the capacity by lower-bounding the mutual information rate for a given set of waveforms with exponentially distributed zero-crossing distances, where we focus on the behavior in the mid to high signal-to-noise ratio regime. We find that in case the input randomness scales appropriately with the available bandwidth, the mutual information rate grows linearly with the channel bandwidth for constant signal-to-noise ratios. Furthermore, for a given bandwidth the lower bound saturates with the signal-to-noise ratio growing to infinity. The ratio between the lower bound on the mutual information rate and the capacity of the additive white Gaussian noise channel without quantization is a constant independent of the channel bandwidth for an appropriately chosen randomness of the channel input and a given signal-to-noise ratio. We complement those findings with an upper bound on the mutual information rate for the specific signaling scheme. We show that both bounds are close in the mid to high SNR domain.